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Chapter 7 | Work, Energy, and Energy Resources 285
it reaches the top of the slope, completely ignoring everything in between. Doing the same type of analysis to find which terms are zero, the conservation of mechanical energy becomes
(7.53) This form of the equation means that the spring’s initial potential energy is converted partly to gravitational potential energy
and partly to kinetic energy. The final speed at the top of the slope will be less than at the bottom. Solving for and substituting known values gives
(7.54)
Discussion
Another way to solve this problem is to realize that the car’s kinetic energy before it goes up the slope is converted partly to potential energy—that is, to take the final conditions in part (a) to be the initial conditions in part (b).
Applying the Science Practices: Potential Energy in a Spring
Suppose you are running an experiment in which two 250 g carts connected by a spring (with spring constant 120 N/m) are run into a solid block, and the compression of the spring is measured. In one run of this experiment, the spring was measured to compress from its rest length of 5.0 cm to a minimum length of 2.0 cm. What was the potential energy stored in this system?
Answer
Note that the change in length of the spring is 3.0 cm. Hence we can apply Equation 7.42 to find that the potential energy is PE = (1/2)(120 N/m)(0.030 m)2 = 0.0541 J.
Note that, for conservative forces, we do not directly calculate the work they do; rather, we consider their effects through their corresponding potential energies, just as we did in Example 7.8. Note also that we do not consider details of the path taken—only the starting and ending points are important (as long as the path is not impossible). This assumption is usually a tremendous simplification, because the path may be complicated and forces may vary along the way.
7.5 Nonconservative Forces
PhET Explorations: Energy Skate Park
Learn about conservation of energy with a skater dude! Build tracks, ramps and jumps for the skater and view the kinetic energy, potential energy and friction as he moves. You can also take the skater to different planets or even space!
Figure 7.13 Energy Skate Park (http://cnx.org/content/m55076/1.5/energy-skate-park_en.jar)
Learning Objectives
By the end of this section, you will be able to:
• Define nonconservative forces and explain how they affect mechanical energy.
• Show how the principle of conservation of energy can be applied by treating the conservative forces in terms of their
potential energies and any nonconservative forces in terms of the work they do.
The information presented in this section supports the following AP® learning objectives and science practices:
• 4.C.1.2 The student is able to predict changes in the total energy of a system due to changes in position and speed of
